hyperbolic metric, triangular ratio metric, Moebius transformation.

51M09; 51M16; 30C20.

Introduction↩︎

In the theory of planar conformal and quasiconformal mappings the hyperbolic metric plays an important role (see, e.g. [1]–[3]). It is invariant under conformal mappings and decreases as the domain expands. Disadvantage of the hyperbolic metric is the fact that it is difficult to calculate in domains with complicated geometry. In Euclidean spaces the class of conformal mappings is narrow and the study of quasiconformal mappings needs the use of other metrics whose properties are close to those of the hyperbolic metric in the plane. Such metrics depends not only on the Euclidean distance between points but also on their location with respect to the boundary of the domain under consideration; they are called intrinsic metrics.

One of such metrics is the triangular ratio metric. For a given domain \(D\subset \mathbb{R}^n\) with a non-degenerate boundary \(\partial D\) the triangular ratio metric \(s_D\) can be defined as follows. If \(u\), \(v\in D\), then \[s_D(u,v)=\frac{|u-v|}{\inf_{w\in \partial D}(|u-w|+|w-v|)}.\] Here \(|u-v|\) denotes the Euclidean distance between \(u\) and \(v\).

The properties of the triangular ratio metric for some classes of domains and its applications in the theory of conformal and quasiconformal mappings were studied in works of J. Chen, P. Hariri, R. Klén, P. Hästö, O. Rainio, M. Vuorinen and others (see, e.g., [4]–[8]).

In the monograph [3] (see also [9]) the problem on distortion of the triangular ratio metric in the unit disk \(\mathbb{U}=\{|z|<1\}\) under Moebius transformation \[\label{moeb} w=f(z)=\frac{z+a}{1+za}, \quad 0\le a<1,\tag{1}\] was stated. It was shown that \[\label{1} s_\mathbb{U}(w_1,w_2)\le L(a)s_\mathbb{U}(z_1,z_2), \quad w_1=f(z_1),\;w_2=f(z_2),\tag{2}\] for all points \(z_1\), \(z_2\in \mathbb{U}\) with a constant \(L(a)\ge 1+a\) and was conjectured that the best value of \(L(a)\) is equal to \(1+a\). We will prove that this conjecture holds.

1 Hyperbolic metric and triangular ratio metric↩︎

First we recall that in the unit disk \(\mathbb{U}\) the hyperbolic metric \(\rho_\mathbb{U}\) can be defined by the equality \[\mathop{\rm th}\nolimits\frac{\rho_\mathbb{U}(u,v)}{2}=\frac{|u-v|}{|1-u\overline{v}|}, \quad u,v\in \mathbb{U}.\] If \(D\) is a simply-connected domain in \(\mathbb{C}\) with nondegenerate boundary and \(f:D\to U\) is a conformal mapping of \(D\) onto \(\mathbb{U}\), then, by definition, the hyperbolic metric \(\rho_D\) in \(D\) is \[\rho_D(u,v)=\rho_\mathbb{U}(f(u),f(v)), \quad u,v\in D.\] This definition is correct, i.e. it does not depend on the choice of the conformal mapping \(f\). As a corollary, we obtain that the hyperbolic metric is invariant under conformal mappings: if \(g:D_1\to D_2\) is a conformal mapping of \(D_1\) onto \(D_2\), then \[\rho_{D_1}(u,v)=\rho_{D_2}(g(u),g(v)), \quad u,v\in D_1.\]

In particular, in the upper half-plane \(\mathbb{H}=\{z\in \mathbb{C}\mid \mathop{\rm Im}\nolimits z>0\}\) we have \[\mathop{\rm th}\nolimits\frac{\rho_\mathbb{H}(u,v)}{2}=\frac{|u-v|}{|u-\overline{v}|}, \quad u,v\in \mathbb{H},\] and in the disk \(K(z_0,R)=\{z\in \mathbb{C} : |z-z_0|<R\}\) \[\label{K} \mathop{\rm th}\nolimits\frac{\rho_{K(z_0,R)}(u,v)}{2}=\frac{R|u-v|}{|R^2-(u-z_0)(\overline{v}-\overline{z}_0)|}, \quad u,v\in K(z_0,R).\tag{3}\] The hyperbolic metric strictly decreases as domain \(D\) expands, i.e. Hyperbolic Metric Principle holds (see e.g. [1]).

In convex domains the triangular ratio metric can be calculated via hyperbolic metrics of supporting half-planes.

As a corollary of Theorem 3 we obtain

2 Proof of the main theorem↩︎

Let \(w_1\), \(w_2\in \mathbb{U}\) and \(G\) be the maximal ellipse with foci at the points \(w_1\) and \(w_2\), the interior of which is in \(\mathbb{U}\). Denote by \(e^{i\phi}\) a point at which the ellipse touches the unit circle. Let \(e^{i\theta}=f^{-1}(e^{i\phi})\). Then, by Theorem 3 and 4 \[\label{sw} s_{\mathbb{U}}(w_1,w_2)=\mathop{\rm th}\nolimits\frac{\rho_{\mathbb{H}_\phi}(w_1,w_2)}{2}\tag{4}\] and \[\label{sz} s_\mathbb{U}(z_1,z_2)\ge \mathop{\rm th}\nolimits\frac{\rho_{\mathbb{H}_\theta}(z_1,z_2)}{2}.\tag{5}\]

Consider the preimage \(D_\phi=f^{-1}(\mathbb{H}_\phi)\) of \(\mathbb{H}_\phi\) under the mapping \(f\). It is bounded by the curve \(\Gamma_{\theta}\) which is the preimage of the supporting line of the unit disk at the point \(e^{i\phi}\). We note that \(\Gamma_{\theta}\) is either a circle or a straight line. Moreover it passes through the point \(f^{-1}(\infty)=-1/a\) and touches the unit circle at the point \(e^{i\theta}\). If \(\Gamma_{\theta}\) is a circle, the cases of internal and external tangency of \(\Gamma_{\theta}\) and \(\partial \mathbb{U}\) are possible (see Fig. 1). Consider two cases.

(a) If \(\Gamma_{\theta}\) is a straight line or a circle with external tangency of \(\Gamma_{\theta}\) we have \(\mathbb{H}_\theta\subset D_\phi\), therefore, because of conformal invariance of hyperbolic metric, by Theorem 2 we have \[\rho_{\mathbb{H}_\phi}(w_1,w_2)=\rho_{{D}_\phi}(z_1,z_2)\le \rho_{\mathbb{H}_\theta}(z_1,z_2),\] and this implies that \[\mathop{\rm th}\nolimits\frac{\rho_{\mathbb{H}_\phi}(w_1,w_2)}{2}\le \mathop{\rm th}\nolimits\frac{\rho_{\mathbb{H}_\theta}(z_1,z_2)}{2}.\] Taking into account 4 and 5 , we obtain \[\label{2a} s_\mathbb{U}(w_1,w_2)\le s_\mathbb{U}(z_1,z_2),\tag{6}\] i.e. in this case we proved a stronger inequality than ?? with the constant \(1\) in the right-hand side, instead of \(1+a\).

(b) It remains to consider the case of internal tangency. In this case, \(\Gamma_{\theta}\) is a circle passing through the points \(-1/a\) and \(e^{i\theta}\) and containing the unit circle in its interior. Let \(z_0\) be the center of \(\Gamma_{\theta}\). Then the origin, as the center of the unit circle, is on the segment with endpoints \(e^{i\theta}\) and \(z_0\). Therefore, \(z_0=-te^{i\theta}\) for some \(t>0\) and the radius of \(\Gamma_{\theta}\) equals \(R=t+1\). Since the point \(-1/a\in \Gamma_{\theta}\), we have \(|z_0+1/a|=R\) or \[\label{R1} |(1-R)e^{i\theta}+1/a|=R.\tag{7}\] Therefore, by the triangle inequality, \(R\ge1/a-(R-1)\), consequently, \[\label{R} R\ge \frac{1}{2}\left(1+\frac{1}{a}\right).\tag{8}\]

Taking into account Theorem 3, we will show that \[\label{a} \mathop{\rm th}\nolimits\frac{\rho_{{D}_\phi}(z_1,z_2)}{2}\le (1+a)s_{\mathbb{U}}(z_1,z_2)=(1+a)\mathop{\rm th}\nolimits\frac{\rho_{\mathbb{H}_\psi}(z_1,z_2)}{2},\tag{9}\] where \(e^{i\psi}\) is the point at which the maximal ellipse with foci at the points \(z_1\) and \(z_2\), contained in \(\mathbb{U}\), touches the unit circle.

By 3 , \[\mathop{\rm th}\nolimits\frac{\rho_{{D}_\phi}(z_1,z_2)}{2}=\frac{R|z_1-z_2|}{|R^2-(z_1-z_0)(\overline{z}_2-\overline{z}_0)|}=\frac{R|z_1-z_2|}{|R^2-(e^{-i\theta}z_1+R-1)(e^{i\theta}\overline{z}_2+R-1)|}\] \[=\frac{R|z_1-z_2|}{|2R-1-(R-1)(e^{-i\theta}z_1+e^{i\theta}\overline{z}_2)-z_1\overline{z}_2|}.\] Besides, \[\mathop{\rm th}\nolimits\frac{\rho_{\mathbb{H}_\psi}(z_1,z_2)}{2}=\frac{|z_1-z_2|}{|2-e^{-i\psi}z_1-e^{i\psi}\overline{z}_2|}\] and from 8 we have \[\label{1a} 1+a\ge\frac{2R}{2R-1}.\tag{10}\]

Taking into account 10 , we see that 9 will be proved if we will show that \[\label{2R} \frac{R|z_1-z_2|}{|2R-1-(R-1)(e^{-i\theta}z_1+e^{i\theta}\overline{z}_2)-z_1\overline{z}_2|}\le \frac{2R}{2R-1}\,\frac{|z_1-z_2|}{|2-e^{-i\psi}z_1-e^{i\psi}\overline{z}_2|}\tag{11}\] or \[\label{b} \left|1-\frac{R-1}{2R-1}\,(e^{-i\theta}z_1+e^{i\theta}\overline{z}_2)-\frac{z_1\overline{z}_2}{2R-1}\right|\ge\frac{1}{2}\,|2-e^{-i\psi}z_1-e^{i\psi}\overline{z}_2|.\tag{12}\]

Denote \(\zeta_k(\varphi)=1-e^{-i\varphi}z_k\), \(k=1\), \(2\). According to Theorem 3 (II), the points \(\zeta_1(\psi)\) and \(\overline{\zeta_2(\psi)}\) have the same argument, consequently, \(|\zeta_1(\psi)+\overline{\zeta_2(\psi)}|=|\zeta_1(\psi)|+|\overline{\zeta_2(\psi)}|\). We rewrite 12 in the form \[\left|\frac{R-1}{2R-1}(2-e^{-i\theta}z_1-e^{i\theta}\overline{z}_2)+\frac{1-z_1\overline{z}_2}{2R-1}\right|\ge\frac{1}{2}\,|2-e^{-i\psi}z_1-e^{i\psi}\overline{z}_2|\] or, taking into account that \[1-z_1\overline{z}_2=1-e^{-i\psi}z_1e^{i\psi}\overline{z}_2=1-(1-\zeta_1(\psi))(1-\overline{\zeta_2(\psi)})=\zeta_1(\psi)+\overline{\zeta_2(\psi)}-\zeta_1(\psi)\overline{\zeta_2(\psi)},\] as \[\label{3} \left|\frac{R-1}{2R-1}(\zeta_1(\theta)+\overline{\zeta_2(\theta)})+\frac{\zeta_1(\psi)+\overline{\zeta_2(\psi)}}{2R-1}-\frac{\zeta_1(\psi)\overline{\zeta_2(\psi)}}{2R-1}\right|\ge\frac{1}{2}\,| \zeta_1(\psi)+\overline{\zeta_2(\psi)}|.\tag{13}\] Denote \(r=| \zeta_1(\psi)+\overline{\zeta_2(\psi)}|\), \[A=\frac{R-1}{2R-1}(\zeta_1(\theta)+\overline{\zeta_2(\theta)})+\frac{\zeta_1(\psi)+\overline{\zeta_2(\psi)}}{2R-1}\,, \quad B=\frac{\zeta_1(\psi)\overline{\zeta_2(\psi)}}{2R-1}\,.\] Since the curve \(z=\zeta_1(\phi)+\overline{\zeta_2(\phi)}\), \(0\le\phi\le 2\pi\), is an ellipse, i.e. convex smooth curve, and the modulus of \(\zeta_1(\phi)+\overline{\zeta_2(\phi)}\) attains its maximal value at the point \(\phi=\psi\), we conclude that the ellipse is outside of the disk of radius \(r\), centered at the origin, and touches its boundary circle at some point \(e^{i\eta}r\). Then the ellipse is in the half-plane \(\{z\in \mathbb{C}\mid \mathop{\rm Re}\nolimits(e^{-i\eta}z)\ge r\}\). This implies \[\mathop{\rm Re}\nolimits(e^{-i\eta}A)=\frac{R-1}{2R-1}\mathop{\rm Re}\nolimits[e^{-i\eta}(\zeta_1(\theta)+\overline{\zeta_2(\theta)})]+\frac{1}{2R-1}\mathop{\rm Re}\nolimits[e^{-i\eta}{\zeta_1(\psi)+\overline{\zeta_2(\psi)}}] \ge \frac{Rr}{2R-1},\] thus, \[\label{c} |A|\ge \frac{Rr}{2R-1}\,.\tag{14}\] At last, since the points \(\zeta_1(\psi)\) and \(\overline{\zeta_2(\psi)}\) have the same argument we obtain \[|\zeta_1(\psi)\overline{\zeta_2(\psi)}|\le \frac{1}{2}\, |\zeta_1(\psi)+\overline{\zeta_2(\psi)}|=\frac{1}{2}\,r,\] therefore, \[\label{d} |B|\le \frac{r}{2(2R-1)}\,.\tag{15}\] From 14 , 15 and the triangle inequality we obtain \[|A-B|\ge|A|-|B|\ge \frac{Rr}{2R-1}\,-\frac{r}{2(2R-1)}\,=\frac{r}{2}\,,\] therefore, we have inequality 13 , equivalent to 9 .

3 Corollaries of the main theorem↩︎

The following corollary improves the statement of Theorem 1.

4 ACKNOWLEDGEMENT↩︎

The author thanks Prof M. Vuorinen for valuable remarks.

5 FUNDING↩︎

This work was financially supported by the Russian Science Foundation (grant No 23–11–00066).

6 CONFLICT OF INTEREST↩︎

The author of this work declare that he has no conflicts of interest.

References↩︎